A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
Figure 6. The curve ABC represents the allowable domain for the continuous solution from the
anode (A) to the cathode (C) of equation (2.39) for real s [97, figure 1].
function of dξ/dx ≡ v,
( ) [ ] 4/5 4
ξmax
25
=
αλv2 + (2α/5)(ζ − γ)v+1− I 2 /IPB
2 4 1/s
λv/(ζ − γ)(1+s) +1
5
ξ
1 − I 2 /IPB
2 ·
4
5 λv/(ζ − γ)(1 − s) − 1 ,
(2.41)
where the boundary condition v = 0atξ = ξ max has been applied. At the electrodes
ξ ≪ ξ max (i.e. T ≪ T max ) and it follows that at the cathode where v → ξ
1 ′ the heat flow
is − 1 2 T 0I[ζ(1+s) + γ(1 − s)] < 0 and negligible compared with the Joule heating. At the
anode the quadratic term in v is dominant, giving
( ) 2 ( ) 4/5 ( )
Tmax ξmax
4αλ dξ 2 ( ) 1 − s 1/s
= →
T 0 ξ 25(1 − I 2 /IPB 2 ) . (2.42)
dx anode 1+s
Inserting this value of (dξ/dx) anode into the overall power balance equation
IV
(1 − I 2 )
= 2 ( ) dξ
5 λT 0I
dx
IPB
2 , (2.43)
anode
which represents Joule heating minus bremsstrahlung being equal to the heat flow to the anode,
a more exact relation than equation (2.26) is found, namely
V
[1 − I 2 ] 1/2 ( ) λ 1/2 ( ) 1+s 1/2s
=
T max , (2.44)
α 1 − s
I 2 PB
where the proportionality of the maximum temperature on the voltage is noted. In the absence
of bremsstrahlung this numerically gives T max (K) = 3.62 × 10 3 V (volt).
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